Luis A. Caffarelli and Avner Friedman

Abstract

An n-dimensional stochastic
process ξ(t) is observed. It is known that ξ(t) has the statistics of an n-dimensional
Brownian motion with any one of possibly n + 1 drifts λ0,⋯,λn (λi are given
n-vectors). We observe the process at a running cost, per unit time, given by ci when
the drift is λi, and after some (stopping) time τ make a decision which hypothesis to
accept; the hypothesis Hj means accepting the drift λj; the drift changes in time in
accordance with a Markov process with n + 1 states and a given transition
probability matrix. The problem of finding the optimal stopping time and optimal
final decision leads to a variational inequality for a degenerate elliptic operator. In
this paper we study this variational inequality and the corresponding free boundary.
We also consider, by purely probabilistic methods, the case where ξ(t) is
k-dimensional, k≠n. The outline of the main results is given at the end of
§2.